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Questions and Answers
What is the value of $x$ in the equation $3x + 7 = 16$?
If the area of a triangle is given by the formula $A = \frac{1}{2}bh$, what is the area when the base $b$ is 10 and the height $h$ is 5?
What is the slope of the line represented by the equation $y = 4x + 2$?
If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $p$ and $q$, what is the value of $pq$?
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What is the derivative of the function $f(x) = x^3 - 3x^2 + 4$?
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Study Notes
Solving Linear Equations
- The equation $3x + 7 = 16$ can be solved for $x$ by isolating the variable.
- Subtract 7 from both sides to obtain $3x = 9$.
- Divide both sides by 3 to get $x = 3$.
Calculating Triangle Area
- The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height.
- Substitute $b = 10$ and $h = 5$ into the formula: $A = \frac{1}{2}(10)(5)$.
- Simplify to get $A = 25$.
Identifying Slope
- The equation $y = 4x + 2$ is in slope-intercept form ($y = mx + c$), where $m$ represents the slope.
- Therefore, the slope of the line is $m = 4$.
Understanding Quadratic Roots
- The product of the roots of a quadratic equation in the form $ax^2 + bx + c = 0$ is given by $c/a$.
- In the equation $x^2 - 5x + 6 = 0$, the product of the roots $p$ and $q$ is $(6)/(1) = 6$.
Finding Derivatives
- The derivative of a function $f(x) = x^n$ is given by $f'(x) = nx^{n-1}$.
- Applying this to the function $f(x) = x^3 - 3x^2 + 4$, we get the derivative $f'(x) = 3x^2 - 6x$.
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Description
Test your understanding of basic algebra concepts with this quiz. You'll tackle linear equations, the area of triangles, slopes of lines, quadratic equations, and derivatives. Challenge yourself and see how well you know these fundamental topics!
